Strong resolving graphs: the realization and the characterization problems

The strong resolving graph $G_{SR}$ of a connected graph $G$ was introduced in [Discrete Applied Mathematics 155 (1) (2007) 356–364] as a tool to study the strong metric dimension of $G$. Basically, it was shown that the problem of finding the strong metric dimension of $G$ can be transformed to the problem of finding the vertex cover number of $G_{SR}$. Since then, several articles dealing with this subject have been published. In this paper, we survey the state of knowledge on the strong resolving graph and also derive some new results.